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Coordinate System for Circular Accelerator
Curverlinear coordinate system
Coordinate system to describe particle motion in an accelerator.
Moves with the particle
Set of unit vectors:
€
ˆ s (s) =d r 0(s)
ds
€
ˆ s
€
ˆ x
€
ˆ y
€
r 0(s)
€
r (s)
€
ˆ x (s) = −ρdˆ s (s)
ds
€
ˆ y (s) = ˆ x (s) × ˆ s (s)
Equation of motion
Equation of motion in transverse plane
€
r (s) = r 0(s) + xˆ x (s) + yˆ y (s)
€
dˆ s (s)ds
= −1ρ
ˆ x (s)
€
dˆ x (s)ds
=1ρ
ˆ s (s)
€
dˆ y (s)ds
= 0
x
x
s
Equation of motion
€
d r (s)dt
=dsdt
[ d r 0ds
+ x' ˆ x + x dˆ x ds
+ y ' ˆ y ] =dsdt
[(1+xρ
)ˆ s + x ' ˆ x + y' ˆ y ]
€
d2 r (s)dt 2 =
dsdt
d v ds≈
v 2
(1+xρ
)2[(x ' '− ρ + x
ρ2 ) ˆ x + x'ρ
ˆ s + y ' ' ˆ y ]
€
v = dsdt
[(1+xρ
)ˆ s + x' ˆ x + y ' ˆ y ] = vsˆ s + vx ˆ x + vy ˆ y
€
v 2 = v =
dsdt[(1+
xρ)2 + x '2 +y '2 ]
Equation of motion
€
d2 r (s)dt 2 ≈
v 2
(1+xρ
)2[(x ' '− ρ + x
ρ2 ) ˆ x + x'ρ
ˆ s + y ' ' ˆ y ] =q v ×
B
γm
€
x' '− ρ + xρ2
= −qBy
γmv(1+
xρ)2
€
y' '= qBx
γmv(1+
xρ)2
€
x' '+ qB'γmv
x = 0
€
y' '− qB'γmv
y = 0
Solution of equation of motion
Comparison with harmonic oscillator: A system with a restoring force which is proportional to the distance from its equilibrium position, i.e. Hooker’s Law:
€
d2x(t)dt 2
+ kx(t) = 0
€
x(t) = Acos( kt + χ)
Where k is the spring constant
€
F =d2x(t)dt 2
= −kx(t)
Equation of motion:
Amplitude of the sinusoidal oscillation
Frequency of the oscillation
transverse motion: betatron oscillation
The general case of equation of motion in an accelerator
€
x' '+kx = 0
€
x(s) = Acos( ks+ χ)
€
x'(s) = −A k sin( ks+ χ)
€
x(s) = Acosh( ks+ χ)
€
x'(s) = −A k sinh( ks+ χ)
For k > 0
Where k can also be negative
For k < 0
Transfer Matrix of a beam transport
The transport matrix from point 1 to point 2 is
€
x(s2)x '(s2)⎛
⎝ ⎜
⎞
⎠ ⎟ =
β2β1(cosψs2s1
+α1 sinψs2s1) β1β2 sinψs2s1
−1+α1α2
β1β2sinψs2s1
+α1 −α2
β1β2cosψs2s1
β1β2(cosψs2s1
−α2 sinψs2s1)
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
x(s1)x '(s1)⎛
⎝ ⎜
⎞
⎠ ⎟
€
x(s) = A βx s( ) cos(ψ(s) + χ)
€
x'(s) = −A 1/βx s( ) sin(ψ(s) + χ) +β 'x s( )2
A 1/βx s( ) cos(ψ(s) + χ)
with:
One Turn Map
Transfer matrix of one orbital turn
€
x(s0 + C)x '(s0 + C)⎛
⎝ ⎜
⎞
⎠ ⎟ =
(cos2πQx +αx,s0sin2πQx ) βx,s0 sin2πQx
−1+αx,s0
2
βx,s0
sin2πQx (cos2πQx −αx,s0sin2πQx )
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
x(s0)x '(s0)⎛
⎝ ⎜
⎞
⎠ ⎟
€
Tr(Ms,s+C ) = 2cos2πQx
€
12Tr(Ms,s+C ) ≤1.0
Closed orbit:
€
x(s+ C)x '(s+ C)⎛
⎝ ⎜
⎞
⎠ ⎟ =
x(s)x'(s)⎛
⎝ ⎜
⎞
⎠ ⎟
€
x(s+ C)x '(s+ C)⎛
⎝ ⎜
⎞
⎠ ⎟ = M(s+ C,s)
x(s)x '(s)⎛
⎝ ⎜
⎞
⎠ ⎟
Stable condition
Stability of transverse motion
Matrix from point 1 to point 2
€
Ms2 |s1= MnM2M1
Stable motion requires each transfer matrix to be stable, i.e. its eigen values are in form of oscillation
€
|M − λI | = 0
€
I =1 00 1⎛
⎝ ⎜
⎞
⎠ ⎟
€
λ2 −Tr M( )λ + det M( ) = 0
€
det M( ) =1
€
λ =12Tr M( ) ±
14[Tr M( )]2 −1
€
12Tr(M) ≤1.0
With and
Focusing from quadrupole
€
x' '+ qB'γm
x = 0
Equation of motion through a quadrupole
For a thin quadrupole, i.e. beam position doesn’t change or x = x, but with a change in slope of the particle’s trajectory, i.e.
€
Δx '= − qB' lγmv
x
€
B = B' x
€
l
€
θ
€
θ
€
ρ =γmvqB
Focusing from quadrupole
Required by Maxwell equation, a single quadrupole has to provide focusing in one plane and defocusing in the other plane €
xf
= −Δx'= qB' lγmv
x
€
1f
=qB' lγmv
x Δx’
f s
Transfer matrix of a qudruploe
Thin lens: length of quadrupole is negligible to the displacement relative to the center of the magnet
€
xx '⎛
⎝ ⎜ ⎞
⎠ ⎟ =
1 0−1f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ xx '⎛
⎝ ⎜ ⎞
⎠ ⎟
€
Δx '= − lρ
= −lqBy
γmv= −
qB' lγmv
x = −kx
Transfer matrix of a drift space
Transfer matrix of a drift space
€
xx '⎛
⎝ ⎜ ⎞
⎠ ⎟ =
1 L0 1⎛
⎝ ⎜
⎞
⎠ ⎟ xx '⎛
⎝ ⎜ ⎞
⎠ ⎟
x Δx’
L s
Lattice
Arrangement of magnets: structure of beam line
Bending dipoles, Quadrupoles, Steering dipoles, Drift space and Other insertion elements
Example: FODO cell: alternating arrangement between focusing and
defocusing quadrupoles
L
One FODO cell
f -f
L
FODO lattice
€
xx '⎛
⎝ ⎜ ⎞
⎠ ⎟ =
1 0−12 f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ 1 L0 1⎛
⎝ ⎜
⎞
⎠ ⎟ 1 01f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1 0−12 f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ 1 L0 1⎛
⎝ ⎜
⎞
⎠ ⎟ xx '⎛
⎝ ⎜ ⎞
⎠ ⎟
=1− 2 L
2
f 22L(1+
Lf)
−2(1− Lf) Lf 2
1− 2 L2
f 2
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
xx'⎛
⎝ ⎜ ⎞
⎠ ⎟
Net effect is focusing Provide focusing in both planes!
Dispersion function
Transverse trajectory is function of particle momentum.
ρ
€
Δθ = θΔpp
ρ+Δρ
€
x = D(s)ΔppDefine
Dispersion function
Momentum spread
Dispersion function
Transverse trajectory is function of particle momentum.
€
x' '− ρ + xρ2
= −qBy
γm(1+
xρ)2
€
By = B0 + B' x
€
x' '+ 1ρ22p0 − p
p+
B'Bρ0
p0p
⎡
⎣ ⎢
⎤
⎦ ⎥ x =
1ρΔpp
€
x = D(s)Δpp
€
D(s+ C) = D(s)
€
D' '+ 1ρ22p0 − p
p+
B'Bρ0
p0p
⎡
⎣ ⎢
⎤
⎦ ⎥ D =
1ρ
Dispersion function: cont’d
In drift space
dispersion function has a constant slope
€
1ρ
= 0
€
B'= 0and
€
⇒ D' '= 0
In dipoles,
€
1ρ≠ 0
€
B'= 0and
€
D' '+[ 1ρ22p0 − p
p]D =
1ρ
Dispersion function: cont’d
For a focusing quad,
dispersion function oscillates sinusoidally €
1ρ
= 0
€
B'> 0and
€
⇒ D' '+B' p0pD = 0
For a defocusing quad,
dispersion function evolves exponentially €
1ρ
= 0
€
B'< 0and
€
⇒ D' '−B' p0pD = 0
Path length and velocity
For a particle with velocity v,
Transition energy : when particles with different energies spend the same time for each orbital turn - Below transition energy: higher energy particle travels faster - Above transition energy: higher energy particle travels slower €
γ t
€
L = vT
€
ΔLL
=Δvv
+ΔTT
€
Δvv
=Δββ
=1γ 2Δpp
€
ΔTT
= (α − 1γ 2)Δpp
= ( 1γ t2 −
1γ 2)Δpp
Compaction factor
The difference of the length of closed orbit between off-momentum particle and on momentum particle, i.e.
€
ΔCC
=αΔpp
=
ρ + DΔpp
⎛
⎝ ⎜
⎞
⎠ ⎟ dθ −∫ ρ∫ dθ
ρ∫ dθ
€
αΔpp
= 〈Dρ〉Δpp⇒α = 〈
Dρ〉
Chromatic effect
Comes from the fact the the focusing effect of an quadrupole is momentum dependent
- Higher energy particle has less focusing
€
1f
=qpkl
Chromaticity: tune spread due to momentum spread
€
ξx,y =ΔQx,y
Δp / p momentum spread
Particles with different momentum have different betatron tune
Tune spread
Chromaticity
Transfer matrix
€
M =1 0−1f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ ≈
1 0−1f(1− Δp
p) 1
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=1 0−1f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1 01fΔpp
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
€
M(s+ C,s) = M(B,A)1 0−1f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
= M(B,A)1 0−1f
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
1 01fΔpp
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
A B
Transfer matrix of a thin quadrupole
Chromaticity
€
M(s+ C,s) =
(cos2πQx +αx,s0sin2πQx ) βx,s0
sin2πQx
−1+αx,s0
2
βx,s0
sin2πQx (cos2πQx −αx,s0sin2πQx )
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
1 01fΔpp
1⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟
=
(cos2πQx +αx,s0sin2πQx ) +
1fΔppβx,s0 sin2πQx βx,s0
sin2πQx
−1+αx,s0
2
βx,s0
sin2πQx + (cos2πQx −αx,s0sin2πQx )
1fΔpp
(cos2πQx −αx,s0sin2πQx )
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
cos[2π (Qx + ΔQx )] =12Tr(M(s+ C,s)
€
cos[2π (Qx + ΔQx )] = cos2πQx +12βx,s0 sin2πQx
1fΔpp
Chromaticity
€
ξx =ΔQx
Δp / p= −
14π
1fβ(s)
€
cos[2π (Qx + ΔQx )] = cos2πQx +12βx,s0 sin2πQx
1fΔpp
Assuming the tune change due to momentum difference is small
€
cos2πQx − 2πΔQx sin2πQx = cos2πQx +12βx,s0
sin2πQx1fΔpp
€
ΔQx = −14π
βx,s01fΔpp
€
ξx =ΔQx
Δp / p= −
14π
kii∑ βx,i
Chromaticity of a FODO cell
€
ξx = −14π
β f1f−βd
1f
⎛
⎝ ⎜
⎞
⎠ ⎟
L
One FODO cell
L
β
βf βd
€
β f ,d =2L(1± sin[Δψ /2])
sin[Δψ]
€
ξx = −1πtanΔψ
2
€
sin[Δψ /2] =Lf
€
ξx = −1π
L / fsinΔψ
Chromaticity correction
Nature chromaticity can be large and can result to large tune spread and get close to resonance condition
Solution: - A special magnet which provides stronger focusing for particles
with higher energy: sextupole
€
Δpp
> 0
€
Δpp
< 0€
Δpp
= 0
Closed orbit distortion
Dipole kicks can cause particle’s trajectory deviate away from the designed orbit - Dipole error - Quadrupole misalignment
€
x(s)x '(s)⎛
⎝ ⎜
⎞
⎠ ⎟ = M(s,s0)[M(s0,s)
x(s)x '(s)⎛
⎝ ⎜
⎞
⎠ ⎟ +
0θ
⎛
⎝ ⎜ ⎞
⎠ ⎟ ]
Assuming a circular ring with a single dipole error, closed orbit then becomes:
s0
BPM
s
Closed orbit: single dipole error
€
x(s0)x '(s0)⎛
⎝ ⎜
⎞
⎠ ⎟ = M(s0 + C,s0)
x(s0)x '(s0)⎛
⎝ ⎜
⎞
⎠ ⎟ +
0θ
⎛
⎝ ⎜ ⎞
⎠ ⎟
Let’s first solve the closed orbit at the location where the dipole error is
€
x(s) = βx (s0)βx (s)θ
2sinπQx
cos ψ(s,s0) −πQx[ ]€
x(s0) = βx (s0)θ
2sinπQx
cosπQx
The closed orbit distortion reaches its maximum at the opposite side of the dipole error location
Closed orbit distortion
In the case of multiple dipole errors distributed around the ring. The closed orbit is
€
x(s) = βx (s) βx (si)i∑ θi
2sinπQx
cos ψ(si,s0) −πQx[ ]
Amplitude of the closed orbit distortion is inversely proportion to sinπQx,y - No stable orbit if tune is integer!
Measure closed orbit
Distribute beam position monitors around ring.
Control closed orbit
minimized the closed orbit distortion. Large closed orbit distortions cause limitation on the
physical aperture Need dipole correctors and beam position monitors
distributed around the ring Assuming we have m beam position monitors and n
dipole correctors, the response at each beam position monitor from the n correctors is:
€
xk = βx,k βx,iθi
2sinπQx
cos ψ(si,s0) −πQx[ ]k=1
n
∑
Control closed orbit
Or,
To cancel the closed orbit measured at all the bpms, the correctors are then
€
θ1θ2θn
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
= M−1( )
x1x2xm
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
€
x1x2xm
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
= M( )
θ1θ2θn
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
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